Correct Answer: $56\%$

Solution : The total surface area of a right circular cylinder $=2\pi r(h + r)$, where $r$ is the radius and $h$ is the height.
We have a height is $18\;\mathrm{cm}$ and a radius is $7\;\mathrm{cm}$.
The initial total surface area $\mathrm{(T_1)}$ of the cylinder,
$\mathrm{T_1}=2\pi \times 7(18 + 7) =350\pi \;\mathrm{cm^2}$
When the cylinder is cut into three equal parts.
The height of each part $=\frac{18}{3} = 6\;\mathrm{cm}$
The total surface area of each part $=2\pi \times7(6 + 7) = 182\pi \;\mathrm{cm^2}$
The final total surface area $\mathrm{(T_2)}$ of all three parts.
$\mathrm{T_2}=3 \times182\pi = 546\pi \;\mathrm{cm^2}$
$\therefore$ The percentage increase $=\frac{\mathrm{T_2}-\mathrm{T_1}}{\mathrm{T_1}} \times 100=\frac{546\pi - 350\pi}{350\pi} \times 100 = 56\%$
Hence, the correct answer is $56\%$.